Jordan trialgebras and post-Jordan algebras

JOURNAL OF ALGEBRA

卷:486

Article

Bagherzadeh, Fatemeh¹  Bremner, Murray¹  Madariag, Sara² 

[1] Univ Saskatchewan, Dept Math & Stat, Saskatoon, SK, Canada

[2] Univ La Rioja, Dept Math & Comp Sci, La Rioja, Spain

关键词: Keywords: Jordan algebras;    Di- and tri-algebras;    Pre- and post-Jordan algebras;    Polynomial identities;    Algebraic operads;    Triplicators;    Trisuccessors;    Koszul duality;    Combinatorics of binary trees;    Computer algebra;    Linear algebra over prime fields;    Representation theory of symmetric groups;   

DOI  :  10.1016/j.jalgebra.2017.04.022

来源: Elsevier

【 摘 要 】

We compute minimal sets of generators for the S-n-modules (n <= 4) of multilinear polynomial identities of arity n satisfied by the Jordan product and the Jordan diproduct (resp. pre-Jordan product) in every triassociative (resp. tridendriform) algebra. These identities define Jordan trialgebras and post-Jordan algebras: Jordan analogues of the Lie trialgebras and post-Lie algebras introduced by Dotsenko et al., Pei et al., Vallette & Loday. We include an extensive review of analogous structures existing in the literature, and their interrelations, in order to identify the gaps filled by our two new varieties of algebras. We use computer algebra (linear algebra over finite fields, representation theory of symmetric groups), to verify in both cases that every polynomial identity of arity <= 6 is a consequence of those of arity <= 4. We conjecture that in both cases the next independent identities have arity 8, imitating the Glennie identities for Jordan algebras. We formulate our results as a commutative square of operad morphisms, which leads to the conjecture that the squares in a much more general class are also commutative. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free